Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Jensen's inequality can fail on an infinite measure space without normalization

Statement refuted

Jensen's inequality remains valid without the hypothesis that the underlying measure be a probability measure.

Facts & Assumptions

Given: Counting measure on N, the function f:=χ{1,2}, and φ(x)=x2.

[L1]

Jensen's theorem is stated for probability measures (Jensen's integral inequality for a probability measure).

[L2]

Counting measure is a measure on N (Counting measure on an arbitrary set, Counting measure is a measure).

Counterexample

technique · direct
1.1

Under counting measure,[L2, given, algebra] ∫f d#=2,∫φ(f) d#=2.

2.1

Hence [step 1.1, L1] ∎ φ ⁣(∫f d#)=4>2=∫φ(f) d#. So Jensen fails on this infinite measure space, exactly as warned by [L1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources